Misc I :i.i. \\T'Ts KXAMT: 

 :nine tlio positions of equilibrium, (1) when the inclina- 

 tion of one of the phu itrr than tan ' , (2) when the 



' 



inclination of both plan.-- is [fl than tan ' . 



1'J. A pack of cards is laid on a table ; each projects in 

 the direction of the length <>f the pack beyond the one hrlow 

 it; if each projects as far as possible, prove that tin- 



.-n the extremities of the successive cards will form an 

 harmonic progression. 



l.'I. Find the least exccntricity of an ellipse in order that 

 it may be capable of resting in equilibrium on a perfectly 

 rougli inclined plane. 



2 sing 

 + sina* 



14. Two mutually repelling particles are placed in a para- 

 bolic groove, and connected by a thread which passes through 

 a small ring at the focus; shew that if the particles br a; 

 either their abscissse arc equal, or the two parts of the t: 

 form one straight line. 



l.~. Kach element of a parabolic arc bounded by the 

 and the latus rectum is acted on by a force in the 

 proportional to the distance of the element from tin- axis of 

 the parabola. Shew that the equation to the straight line 

 in which the resultant acts is 



1 :// -f 10x = 2Ga. 



l'. Each element of the arc of an elliptic quadra; 

 acted on by a force in the normal proportional to the, ordinate 

 of that point. Shew that the equation to the straight line 

 in which the resultant acts is 



6 by - ZTTOX + 4a* - 4i* = 0. 



17. A smooth body in the form of a sphere is divided into 

 hemispheres and placed with the plane of division vertical 



ntal plane; a strini: I'-aded at it 



tremities with two equal weights han^s upon the sphere, 

 passing over its highest point and cutting the plane of dn 



